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Finding the Nth-Term in the Growing Squares Problem Before reading the descriptions of the board work try to figure out what you can about each teacher's approach. Kevin Kevin noticed that the second figure had 3 horizontal rows of 2 toothpicks and 3 vertical columns of 2 toothpicks. That is two groups of 6, or 12 toothpicks total. Then he saw that the same was true for Figure 3. There were 4 horizontal rows of 3 toothpicks and 4 vertical columns of 3 toothpicks, or 2 groups of 12 toothpicks, or 24 total toothpicks. He saw that the same was true for Figure 4. There were 5 horizontal rows of 4 toothpicks and 5 vertical columns of 4 toothpicks, or 2 groups of 20 toothpicks, or 40 total toothpicks. Kevin (continued) Given the Nth figure, Kevin realized he would have 2 groups of N(N+1) toothpicks. Eric 2n2 + 2n is 4 times the formula for calculating the triangular numbers (n2 + n)/2 Jane Jane noticed the 4th figure has 4 rows of 4 horizontal toothpicks (pink), and 4 rows of 4 vertical toothpicks (green) or 2(4)2 toothpicks. Then she saw that there were 2 rows of 4 toothpicks remaining (grey), one making up the right side of the square and the other making up the bottom. The same was true for the 5th figure and Jane was able to come up with the same generalization as Eric, 2x2 + 2x. Cynthia Cynthia saw that the number of toothpicks followed the pattern of 4 times the sequence of triangular numbers. So for example, since the 2nd triangular number is 3, the # of toothpicks in the 2nd figure is 4(3). Since the 3rd triangular number is 6, the # of toothpicks in the 3rd figure would be 4(6). As with Eric and Jane's formula, Cynthia's is 4 times the formula for calculating the triangular numbers - (n2 + n)/2 4(n(n+1)) 4(n2 +n) 2 2 Mark, Charlie, Ramon After finding the iterative rule for figuring out the number of squares and the number of toothpicks in each figure, we got interested (a.k.a. Mark got fixated) in the idea of visualizing separating each square in a figure and trying to figure out the pattern in how many toothpicks are duplicated (and need to be removed), when the squares are put back together. For example, the 2nd figure is made up of 4 squares. If we separate those 4 squares, it would take 16 toothpicks to construct them. But when we put them back together, we need to remove 4, leaving the 2nd figure with only 12 toothpicks. If we separate the 9 squares in the 3rd figure, it would be 36 toothpicks. But when we put them back together we need to subtract 12 toothpicks, leaving us with 24. When time ran out we were trying to find the relationship between the figure number and the number of toothpicks we had to subtracted when we brought the squares back together. Figure Number (x) 4(Figure Difference between Figure Number 4(Figure Number Number # of Squared) Squared) and the actual # Squared toothpicks of toothpicks in the x2 figure 4(x)2 1 1 4 4 0 1(0) 2 4 12 16 4 2(2) 3 9 24 36 12 3(4) 4 16 40 64 24 4(6) 5 25 60 100 40 5(8) 6 36 84 144 60 6(10) 4x2 - ___________ The generalization to the left should read: n2 + ((n+1)2 - 1) Charlie came up with the idea by looking at the relationship between the number of squares and the number of toothpicks. He noticed in Fig. 1, there were 4 toothpicks and 1 square - a difference of 3. He noticed that figure 2 had 12 toothpicks and 4 squares a difference of 8. He saw that Fig 3 had 24 toothpicks and 9 squares - a difference of 15. Then he realized the the difference between the # of toothpicks and the number of squares for each figure was one less than the number of squares in the next figure. So for example, the difference between the number of toothpicks and the number of squares in the 5th figure would be 35, because that is one less than the square of the next figure (62 = 36). Since we know the 5th figure has 25 squares, we know it would have 25 + (36-1) toothpicks, which it does... 60 toothpicks. But in testing the generalization as it is written in the photo, I realized it is what was missing from the generalization we were trying to build above when we separated and then recombined the squares... 4x2 - (n2 + (n-1)2 - 1), Jane's Growing Triangles In the growing triangles pattern (Fig 4 is the one drawn above), Jane noticed that the same thing that happens to the vertical toothpicks, happens to the toothpicks on both diagonals - three times in all. That is to say the outside edge of each triangle is the figure number, and then the number of toothpicks goes down by one, all the way to one toothpick. For example - in the 4th figure above, the right side of the triangle has 4 black toothpicks, followed by 3, then 2, then 1. The same is true for the sea foam green and the fern green. This was a pattern she recognized from Gauss and the Handshake problem. This formula is 3 times the formula for calculating triangular numbers: (n2 +n)/2